Hence, at no-load or zero current, El = E0, decreases
with increasing load, reaches a minimum at OE^ perpen-
dicular to clt and then increases again, reaches once more
Fig. 146.
El = EQ at E?, and then increases beyond E0. The cur-
rent is always ahead of the induced E.M.F. El of the motor,
and by its lead compensates for the self-induction of the
system, making the total circuit non-inductive.
The power is a maximum at Ef, where OEf = EfEQ =
1/2 x ~OE^ and is then = / x "^7/2. Hence, since OEf =
EJ2,f=E()/2randP
hence = the maxi-
mum power which, over a non-inductive line of resistance r
can be transmitted, at 50 per cent, efficiency, into a non-
inductive circuit.
-334 ALTERNATING-CURRENT PHENOMENA.
In this case,
In general, it is, taken from the diagram, at the condi-
tion of maximum efficiency :
Comparing these results with those in Chapter IX. on
Self-induction and Capacity, we see that the condition of
maximum efficiency of the synchronous motor system is
the same as in a system containing only inductance and
•capacity, the lead of the current against the induced E.M.F.
El here acting in the same way as the condenser capacity
in Chapter IX.
Fig. 147.
D. En = constant ; P = constant.
If the power of a synchronous motor remains constant,
we have (Fig. 147) / x OE^ = constant, or, since OE1 —
SYNCHRONOUS MOTOR.
335
Ir, I = OE1/ r, and: OE1 x OE? = O£l X E1EJ =
constant.
Hence we get the diagram for any value of the current
/, at constant power Plt by making OE1 = I r, E1E01 = Pl j I
erecting in EQl a perpendicular, which gives two points of
intersection with circle eQ, EQ, one leading, the other lagging.
Hence, at a given impressed E.M.F. EQ, the same power P±
E,
1250 7
1100/1580 31/16.7
1480 32
1050/1840 2/25
2120
2170
37.5
40
45.5
16.7
Fig. U8.
can be transmitted by the same current I with two different
induced E.M.Fs. E} of the motor; one, OEl = EEQ small,
corresponding to a lagging current ; and the other, OEl =
EEQ large, corresponding to a leading current. The former
is shown in dotted lines, the latter in drawn lines, in the
diagram, Fig. 147.
Hence a synchronous motor can work with a given out-
put, at the same current with two different counter E.M.Fs.
336
ALTERNATING-CURRENT PHENOMENA.
E1. In one of the cases the current is leading, in the
Dther lagging.
In Figs. 148 to 151 are shown diagrams, giving the points
E0 = impressed E.M.F., assumed as constant = 1000 volts,
E = E.M.F. consumed by impedance,
E' = E.M.F. consumed by resistance.
EflOOO
P=6000
34O< E,<1920
7< I < 43
Fig. 149.
I
1450 17.3
1170/1910 10/30
1040/1930 8/37.5
10/30
17.3
of the motor, Elt is OElt equal and
shown in the diagrams, to avoid
The counter E.M.F.
parallel EEQ, but not
complication.
The four diagrams correspond to the values of power,
or motor output,
P = 1,000, 6,000,
9,000,
12,000 watts, and give :
1 < I < 49 Fig. 132.
P = 1,000 46 < El < 2,200,
P = 6,000 340 < £, < 1,920, 7 < I < 43 Fig. 133.
P = 9,000 540 < El < 1,750, 11.8 < / < 38.2 Fig. 134.
P = 12,000 920 < El < 1,320, 20 < I < 30 Fig. 153.
SYNCHRONOUS MOTOR.
337
E, I
3 1200/1660 15/30
1080/1750 13/34.7
900/1590 11.8/38.2.
720/1100 13/34.7
620/820 15/30
/3 540 21.2
3 1280 24.5
2 1120/1320 21/28.6
all— l-QQO/1260 30/30
920/1100
020
21/28.6
24.5
P=I200O
920< E,< 1320
20<l<30
Fig. 151.
As seen, the permissible value of counter E.M.F. Ev and
of current /, becomes narrower with increasing output.
338 ALTERNATING-CURRENT PHENOMENA.
In the diagrams, different points of EQ are marked with
1, 2, 3 . . . , when corresponding to leading current, with
21, 31, . . . , when corresponding to lagging current.
The values of counter E.M.F. Ev and of current 7 are
noted on the diagrams, opposite to the corresponding points
*o-
In this condition it is interesting to plot the current as
function of the induced E.M.F. El of the motor, for con-
stant power /V Such curves are given in Fig. 155 and
explained in the following on page 345.
- While the graphic method is very convenient to
get a clear insight into the interdependence of the different
quantities, for numerical calculation it is preferable to ex-
press the diagrams analytically.
For this purpose,
Let z = Vr2 -j- x2 = impedance of the circuit of (equivalent)
resistance r and (equivalent) reactance x = 2 TT NL, containing
the impressed E.M.F. e0* and the counter E.M.F. et of the syn-
chronous motor; that is, the E.M.F. induced in the motor arma-
ture by its rotation through the (resultant) magnetic field.
Let i = current in the circuit (effective values).
The mechanical power delivered by the synchronous
motor (including friction and core loss) is the electric
power consumed by the C. E.M.F. e1; hence —
p = *>! cos ft,^), (1)
thus, —
- If f0 = E.M.F. at motor terminals, z = internal impedance of the
motor; if eo= terminal voltage of the generator, z = total impedance of line
and motor; if t0= E.M.F. of generator, that is, E.M.F. induced in generator
armature by its rotation through the magnetic field, z includes the generator
impedance also.
SYNCHRONOUS MOTOR. 339
The displacement of phase between current i and E.M.F.
= z i consumed by the impedance z is :
cos (ie) = -
sin (/<?)
x
(3)
Since the three E.M.Fs. acting in the closed circuit :
e0 = E.M.F. of generator,
fi = C.E.M.F. of synchronous motor,
e = zi = E.M.F. consumed by impedance,
form a triangle, that is, c^ and e are components of ^0, it is
(Fig. 152) :
e1 „ 2 eZ .1 „?. ^2 ,'2
hence, cos (,.#) = •- — — = -0 — - — . (5)
2 e^e '2,zie^
since, however, by diagram :
cos (el , e) = cos (/, e — /', e^)
= cos (/, e) cos (/, ^i) + sin (t, e) sin (/, ^) (6)
substitution of (2), (3) and (5) in (6) gives, after some trans-
position :
the Fundamental Equation of tJie Synchronous Motor, relat-
ing impressed E.M.F., <?0 ; C. E.M.F., ^ ; current z; power,
/, and resistance, r ; reactance, x ; impedance s.
This equation shows that, at given impressed E.M.F. e$f
and given impedance s = Vr2 + x*, three variables are left,
ev i,p, of which two are independent. Hence, at given ^
and s, the current i is not determined by the load / only,
but also by the excitation, and thus the same current i can
represent widely different loads p, according to the excita-
tion ; and with the same load, the current i can be varied
in a wide range, by varying the field excitation e1.
The meaning of equation (7) is made more perspicuous
340 ALTERNATING-CURRENT PHENOMENA.
by some transformations, which separate ev and i, as func-
tion of/ and of an angular parameter <£.
Substituting in (7) the new coordinates :
V2
V2
or,
_
V2
we get
substituting again, e<f = a
Izp = b
r = €Z
hence, x = z Vl — e2
jr. 753.
we jret
a — a V2 — e b = V(l — e2) (2 a2 — 2 £2 -
and, squared,
substituting
gives, after some transposition,
v* -f ze/2 = (-1 ~ *") a (a — 2 tb\
(9)
)» (11)
— 0, (12)
(13)
(14)
SYNCHRONOUS MOTOR. 341
hence'if
i* + w* = £* (16)
the equation of a circle with radius R.
Substituting now backwards, we get, with some trans-
positions :
{r* (ef + zi2) - z (Vo2 - 2 r/)}2 + {r x (e? - z*i2)}2 =
*2.sV(^02-4r/) (17)
the Fundamental Rquation of the Synchronous Motor in a
modified form.
The separation of e± and i can be effected by the intro-
duction of a parameter <£ by the equations :
r3- (e? — z2 /2) - z2 (ef — 2rp)=xze() V<r0a — ±rp cos <£
rx (e? - z2/2) =xze» Vtf - 4 r/ sin ' l '
These equations (18), transposed, give
N
The parameter <^> has no direct physical meaning, appar-
ently.
These equations (19) and (20), by giving the values ef
el and i as functions of / and the parameter <£ enable us
to construct the Power Characteristics of the Synchronous
Motor, as the curves relating ev and i, for a given power /,
by attributing to <£ all different values.
342 ALTERNATING-CURRENT PHENOMENA.
Since the variables v and w in the equation of the circle
(16) are quadratic functions of e1 and /', the Power Charac-
teristics of the Synchronous Motor are Quartic Curves.
They represent the action of the synchronous motor
under all conditions of load and excitation, as an element
of power transmission even including the line, etc.
Before discussing further these Power Characteristics,
some special conditions may be considered.
- A. Maximum Output.
Since the expression of el and i [equations (19) and
(20)] contain the square root, W02 — 4 rp, it is obvious
that the maximum value of / corresponds to the moment
where this square root disappears by passing from real to
imaginary ; that is,
tf _ 4 rp = 0,
°r>
/ = £.. (21)
This is the same value which represents the maximum
power transmissible by E.M.F., eQ, over a non-inductive line
of resistance, r\ or, more generally, the maximum power
which can be transmitted over a line of impedance,
into any circuit, shunted by a condenser of suitable capacity.
Substituting (21) in (19) and (20), we get,
and the displacement of phase in the synchronous motor.
cor(A,0-^--i
tc± z
hence,
tan fa, /) = -?, (23)
SYNCHRONOUS MOTOR. 343
that is, the angle of internal displacement in the synchron-
ous motor i§ equal, but opposite to, the angle of displace-
ment of line impedance,
('i, 0 = - (', 0,
= ~ <X '), (24)
and consequently,
(.-0,0=0; (25)
that is, the current, z, is in phase with the impressed
E.M.F., *0.
If 2 < 2 r, el < <?0; that is, motor E.M.F. < generator E.M.F.
If z = 2 r, el = e0 ; that is, motor E.M.F. = generator E.M.F.
If z > 2 r, <?! > r0; that is, motor E.M.F. > generator E.M.F.
In either case, the current in the synchronous motor is
leading.
- B. Running Light, p = 0.
When running light, or for / = 0, we get, by substitut-
ing in (19) and (20),
(26)
Obviously this condition cannot well be fulfilled, since p
must at least equal the power consumed by friction, etc. ;
and thus the true no-load curve merely approaches the curve
/ = 0, being, however, rounded off, where curve (26) gives
sharp corners.
Substituting / = 0 into equation (7) gives, after squar-
ing and transposing,
e* + e<* 4- 3*,-« - 2 ^V - 2 22rV + 2 ra*'V - 2 2V = 0. (27)
This quartic equation can be resolved into the product
of two quadratic equations,
-
| (28)
-
j
344 ALTERNATING-CURRENT PHENOMENA.
which are the equations of two ellipses, the one the image
of the other, both inclined with their axes.
The minimum value of C.E.M.F., eit is ^ = 0 at / = ^2. (29)
The minimum value of current, z, is / = 0 at et = e0 . (30)
The maximum value of E.M.F., elt is given by Equation (28)',
/= e* + 22z2 -e<?±2 xiel = 0 ;
by the condition,
hence,
The maximum value of current, z, is given by equation
(28) by
— = 0, as
del
(32)
If, as abscissas, elt and as ordinates, zi, are chosen, the
axis of these ellipses pass through the points of maximum
power given by equation (22).
It is obvious thus, that in the V-shaped curves of syn-
chronous motors running light, the two sides of the curves
are not straight lines, as usually assumed, but arcs of ellipses,
the one of concave, the other of convex, curvature.
These two ellipses are shown in Fig. 154, and divide the
whole space into six parts — the two parts A and A', whose
areas contain the quartic curves (19) (20) of synchronous
motor, the two parts B and B', whose areas contain the
quartic curves of generator, and the interior space C and
exterior space D, whose points do not represent any actual
condition of the alternator circuit, but make el , i imaginary.
A and A' and the same B and B' ', are identical condi-
tions of the alternator circuit, differing merely by a simul-
SYNCHRONOUS MOTOR.
345
\
r
\
I
\
4000 3000 ^ 2000 1000
Volts 1000 2000/3000 4000 5000
\
/A'
\
\
Fig. 154.
taneous reversal of current and E.M.F. ; that is, differing
by the time of a half period.
Each of the spaces A and B contains one point of equa-
tion (22), representing the condition of maximum output
of generator, viz., synchronous motor.
- C. Minimum Current at Given Power.
The condition of minimum current, t, at given power, /,
is determined by the absence of a phase displacement at the
impressed E.M.F. eQ,
346 AL TERNA TING-CURRENT PHENOMENA.
This gives from diagram Fig. 153,
e1* = e(? + iz-2ie0r, (33)
or, transposed,
This quadratic curve passes through the point of zero
current and zero power,
through the point of maximum power (22),
and through the point of maximum current and zero power,
enx
r
(35)
and divides each of the quartic curves or power character-
istics into two sections, one with leading, the other with
lagging, current, which sections are separated by the two
points of equation 34, the one corresponding to minimum,
the other to maximum, current.
It is interesting to note that at the latter point the
current can be many times larger than the current which
would pass through the motor while at rest, which latter
current is,
/ = 'J2, (36)
while at no-load, the current can reach the maximum value,
/=^, (35)
the same value as would exist in a non-inductive circuit of
the same resistance.
The minimum value at C.E.M.F. el} at which coincidence
SYNCHRONOUS MOTOR. 347
of phase (eQ , -i) = 0, can still be reached, is determined from
equation (34) by,
as
i — e - — - (37}
The curve of no-displacement, or of minimum current, is
shown in Figs. 138 and 139 in dotted lines.*
-
D. Maximum Displacement of Phase.
(e%, i} = maximum.
At a given power/ the input is,
A =P + i*r = e,i cos (*0, *) ; (38)
hence,
cosfo, 0 = /+/V. (39)
At a given power /, this value, as function of the current
i, is a maximum when
d_(p +
di\
this gives,
(40)
or,
(41)
That is, the displacement of phase, lead or lag, is a
maximum, when the power of the motor equals the power
- It is interesting to note that the equation (34) is similar to the value,
<?! = \/(^0 — 2 r)2 — z'2jr2, which represents the output transmitted over an
inductive line of impedance, z = vV2 + jr2 into a non-inductive circuit.
Equation (34) is identical with the equation giving the maximum voltage,
e± , at current, i, which can be produced by shunting the receiving circuit with a
condenser; that is, the condition of " complete resonance " of the line, z =
x
Vr'2 + x'2, with current, ». Hence, referring to equation (35), el = t0 ~ is
the maximum resonance voltage of the line, reached when closed by a con-
denser of reactance, — x.
348
ALTERNATING-CURRENT PHENOMENA.
consumed by the resistance ; that is, at the electrical effi-
ciency of 50 per cent.
Substituting (40) in equation (7) gives, after squaring
/ N
TSOO 8000^ #WU 3000 3uOO
Fig. 155.
and transposing, the Ouartic Equation of Maximum Dis-
placement,
<>02 - e*y + **z2 (s2 + 8 r2) + 2 j*e* (5 r2 - 22) - 2 / V
(32 + 3 ^ = Oi (42)
The curve of maximum displacement is shown in dash-
dotted lines in Figs. 154 and 155. It passes through the
SYNCHRONOUS MOTOR. 349
point of zero current — as singular or nodal point — and
through the point of maximum power, where the maximum
displacement is zero, and it intersects the curve of zero
displacement.
210. E. Constant Counter E.M.F.
At constant C.E.M.F., el = constant,
If
the current at no-load is not a minimum, and is lagging.
With increasing load, the lag decreases, reaches a mini-
mum, and then increases again, until the motor falls out of
step, without ever coming into coincidence of phase.
If
the current is lagging at no load ; with increasing load the
lag decreases, the current comes into coincidence of phase
with eQ , then becomes leading, reaches a maximum lead ;
then the lead decreases again, the current comes again into
coincidence of phase, and becomes lagging, until the motor
falls out of step.
If eQ < <?! , the current is leading at no load, and the
lead first increases, reaches a maximum, then decreases ;
and whether the current ever comes into coincidence of
phase, and then becomes lagging, or whether the motor
falls out of step while the current is still leading, depends,
whether the C.E.M.F. at the point of maximum output is
> <?0 or < *0.
211. F. Numerical Instance.
Figs. 154 and 155 show the characteristics of a 100-
kilowatt motor, supplied from a 2500-volt generator over a
distance of 5 miles, the line consisting of two wires, No.
2 B. & S.G., 18 inches apart.
350 ALTERNATING-CURRENT PHENOMENA.
In this case we have,
<?0 = 2500 volts constant at generator terminals; ^|
r — 10 ohms, including line and motor ; /^gs
x = 20 ohms, including line and motor ; j
hence z = 22.36 ohms.
Substituting these values, we get,
25002 - e* - 500 i* - 20 / = 40 V*V -/2 (7)
{^2 + 500 ?2 - 31.25 X 106 + 100 /}2 + (2 ^2 - 1000 /2}2 =
7.8125 x 1015 - 5 + 109/. (17)
el = 5590 (19)
V| {(1 — 3.2 x 10~6/) + (.894 cos <£+ .447sin <£) Vl-6.4xlO-6/}.
* = 559 (20)
— 6.4xlO-6/}.
Maximum output,
p = 156.25 kilowatts (21)
at *i = 2,795 volts
i = 125 amperes
Running light,
^ + 500 /a - 6.25 x 104 =p 40 /^ = 0
^ = 20 / ± V6.25 X 104 — 100 i*
At the" minimum value of C.E.M.F. e1 = 0 is / = 112 (29)
At the minimum value of current, / = 0 is el = 2500 (30)
At the maximum value of C.E.M.F. ev = 5590 is / = 223.5 (31)
At the maximum value of current i — 250 is el = 5000 (32)
Curve of zero displacement of phase,
€l = 10 V(250 - O2 + 4 *a (34)
= 10 V6.25 x 104 — 500 / + 5 / 2
Minimum C.E.M.F. point of this curve,
/ = 50 ^ = 2240 (35)
Curve of maximum displacement of phase,
/ = 10 *'2 (40)
(6.25 X 106-^2)2 + .65 X 106 /« - 1010/2 = 0. (42)
SYNCHRONOUS MOTOR. 351
Fig. 154 gives the two ellipses of zero power, in drawn
lines, with the curves of zero displacement in dotted, the
curves of maximum displacement in dash-dotted lines, and
the points of maximum power as crosses.
Fig. 155 gives the motor-power characteristics, for,
/ = 10 kilowatts.
p = 50 kilowatts.
/ = 100 kilowatts.
p = 150 kilowatts.
p = 156.25 kilowatts.
together with the curves of zero displacement, and of maxi-
mum displacement.
212. G. Discussion of Results.
The characteristic curves of the synchronous motor, as
shown in Fig. 155, have been observed frequently, with
their essential features, the V-shaped curve of no load, with
the point rounded off and the two legs slightly curved, the
one concave, the other convex ; the increased rounding off
and contraction of the curves with increasing load ; and
the gradual shifting of the point of minimum current with
increasing load, first towards lower, then towards higher,
values of C.E.M.F. el.
The upper parts of the curves, however, I have never
been able to observe experimentally, and consider it as
probable that they correspond to a condition of synchro-
nous motor-running, which is unstable. The experimental
observations usually extend about over that part of the
curves of Fig. 155 which is reproduced in Fig. 156, and in
trying to extend the curves further to either side, the motor
is thrown out of synchronism.
It must be understood, however, that these power char-
acteristics of the synchronous motor in Fig. 155 can be con-
sidered as approximations only, since a number of assump-
352
ALTERNA TING-CURRENT PHENOMENA.
tions are made which are not, or only partly, fulfilled in
practice. The foremost of these are : •
1. It is assumed that el can be varied unrestrictedly,
while in reality the possible increase of el is limited by
magnetic saturation. Thus in Fig. 155, at an impressed
E.M.F., eQ = 2,500 volts, el rises up to 5,590 volts, which
may or may not be beyond that which can be produced
by the motor, but certainly is beyond that which can be
constantly given by the motor.
Fig. 156.
2. The reactance, x, is assumed as constant. While
the reactance of the line is practically constant, that of the
motor is not, but varies more or less with the saturation,
decreasing for higher values. This decrease of x increases
the current /, corresponding to higher values of elt and
thereby bends the curves upwards at a lower value of ^
than represented in Fig. 155.
It must be understood that the motor reactance is not
a simple quantity, but represents the combined effect of
SYNCHRONOUS MOTOR. 353
self-induction, that is, the E.M.F. induced in the armature
conductor by the current flowing therein and armature
reaction, or the variation of the C. E.M.F. of the motor
by the change of the resultant field, due to the superposi-
tion of the M.M.F. of the armature current upon the field
excitation ; that is, it is the " synchronous reactance."
3. These curves in Fig. 155 represent the conditions
of constant electric power of the motor, thus including the
mechanical and the magnetic friction (core loss). While
the mechanical friction can be considered as approximately
constant, the magnetic friction is not, but increases with
the magnetic induction ; that is, with elf and the same holds
for the power consumed for field excitation.
Hence the useful mechanical output of the motor will
on the same curve, / = const., be larger at points of lower
C.E.M.F., elt than at points of higher e^\ and if the curves
are plotted for constant useful mechanical output, the whole
system of curves will be shifted somewhat towards lower
values of ^ ; hence the points of maximum output of the
motor correspond to a lower E.M.F. also.
It is obvious that the -true mechanical power-character-
istics of the synchronous motor can be determined only
in the case of the particular conditions of the installation
under consideration.
354 AL TERN A TING-CURRENT PHENOMENA,
CHAPTER XX.
COMMUTATOR MOTORS.
213. Commutator motors — that is, motors in which
the current enters or leaves the armature over brushes
through a segmental commutator — have been built of
various types, but have not found any extensive appli-
cation, in consequence of the superiority of the induction
and synchronous motors, due to the absence of commu-
tators.
The main subdivisions of commutator motcrs are the
repulsion motor, the series motor, and the shunt motor.
REPULSION MOTOR.
214. The repulsion motor -is an induction motor or
transformer motor ; that is, a motor in which the main
current enters the primary member or field only, while
in the secondary member, or armature, a current is in-
duced, arid thus the action is due to the repulsive thrust
between induced current and inducing magnetism.
As stated under the heading of induction motors, a
multiple circuit armature is required for the purpose of
having always secondary circuits in inductive relation to
the primary circuit during the rotation. If with a single-
coil field, these secondary circuits are constantly closed
upon themselves as in the induction motor, the primary
circuit will not exert a rotary effect upon the armature
while at rest, since in half of the armature coils the cur-
rent is induced so as to give a rotary effort in the one
direction, and in the other half the current is induced to
COMMUTATOR MOTORS.
355
give a rotary effort in the opposite direction, as shown
by the arrows in Fig. 157.
In the induction motor a second magnetic field is used
to act upon the currents induced by the first, or inducing
magnetic field, and thereby cause a rotation. That means
the motor consists of a primary electric circuit, inducing
Fig. 157.
in the armature the secondary currents, and a primary
magnetizing circuit producing the magnetism to act upon
the secondary currents.
In the polyphase induction motor both functions of the
primary circuit are usually combined in the same coils ; that
is, each primary coil induces secondary currents, and pro-
duces magnetic flux acting upon secondary currents induced
by another primary coil.
356
AL TERNA TING-CURRENT PHENOMENA.
215. In the repulsion motor the difficulty due to the
equal and opposite rotary efforts, caused by the induced
armature currents when acted upon by the inducing mag-
netic field, is overcome by having the armature coils closed
upon themselves, either on short circuit or through resist-
ance, only in that position where the induced currents give
Fig. 158.
a rotary effort in the desired direction, while the armature
coils are open-circuited in the position where the rotary
effort of the induced currents would be in opposition to
the desired rotation. This requires means to open or close
the circuit of the armature coils and thereby introduces the
commutator.
Thus the general construction of a repulsion motor is
as shown in Figs. 158 and 159 diagrammatically as bipolar
COMMUTATOR MOTORS.
357
motor. The field is a single-phase alternating field F, the
armature shown diagrammatically as ring wound A consists
of a number of coils connected to a segmental commutator
C, in general in the same way as in continuous-current ma-
chines. Brushes standing under an angle of about 45° with
the direction of the magnetic field, short-circuit either a
Fig. 159.
part of the armature coils as shown in Fig. 158, or the
whole armature by a connection from brush to brush as
shown in Fig. 159.
The former arrangement has the disadvantage of using a
part of the armature coils only. The second arrangement
has the disadvantage that, in the passage of the brush from
segment to segment, individual armature coils are short-
358
AL TERNA TING-CURRENT PHENOMENA.
circuited, and thereby give a torque in opposite direction to
the torque developed by the main induced current flowing
through the whole armature from brush to brush.
216. Thus the repulsion motor consists of a primary
electric circuit, a magnetic circuit interlinked therewith,
and a secondary circuit closed upon itself and displaced in
Fig. 160.
space by 45° — in a bipolar motor — from the direction of
the magnetic flux, as shown diagrammatically in Fig. 160. *
This secondary circuit, while set in motion, still remains
in the same position of 45° displacement, with the magnetic
flux, or rather, what is theoretically the same, when moving
out of this position, is replaced by other secondary circuits
entering this position of 45° displacement.
For simplicity, in the following all the secondary quan-
COMMUTATOR MOTORS. 359
titles, as E.M.F., current, resistance, reactance, etc., are
assumed as reduced to the primary circuit by the ratio of
turns, in the same way as done in the chapter on Induction
Motors.
217. Let
$ = maximum magnetic flux per field pole ;
e = effective E.M.F. induced thereby in the field turns ; thus,
where ;/ = number of turns, N= frequency.
<?108
thus, 4> = — --
\&-anN
The instantaneous value of magnetism is
<f> = <& sin (3 ;
and the flux interlinked with the armature circuit
<£x = <I> sin /3 sin X ;
when X is the angle between the plane of the armature coil
and the direction of the magnetic flux. (Usually about 45°.)
The E.M.F. induced in the armature circuit, of n turns,
(as reduced to primary circuit), is thus,
e = _ n ^1 10-8, = - n® 4- sin B sin X lO"8,
at at
= - n$> sin X cos (3 + sin (3 cos X 10~8.
If N= frequency in cycles per second, N: = frequency
of rotation or speed in cycles per second, and k = N^/ N
speed
we have
frequency
thus, gl = — 2-TrnJV® {sin X cos /? + k cos X sin B\ 10~8,
or, since $ = — — — — ,
et = e V2 {sin X cos /3 + k cos X sin fi\.
360 ALTERNATING-CURRENT PHENOMENA.
218. Introducing now complex quantities, and counting
the time from the zero value of rising magnetism, the mag-
netism is represented by /4>,
the primary induced E.M.F., E = — e,
the secondary induced E.M.F., £1 = — e {sin X +j"k cos X|;
hence, if
Zl = r1—jx1= secondary impedance reduced to primary circuit,
Z = r — jx = primary impedance,
Y = g —jb = exciting admittance,
we have,
& sin X -f- jk cos A
secondary current, 7X = — L = - e - _ - ,
primary exciting current, I0 = eY= e (g +jb},
hence, total primary current,
Primary impressed E.M.F., E0= — E + IZ\
= e 1 + (sinX
Neglecting in E0 the last term, as of higher order,
£0 = e j 1 + sin X +jk cos X ^ ^4^ j ;
or, eliminating imaginary quantities,
e V(?i + r sin X -f- kx cos X)2 + (x^ + x sin X — kr cos X)2
The power consumed by the component of primary
counter E.M.F., whose flux is interlinked with the secondary
e sin X, is,
f = [e sin X /]' = ^inXfosuiX-^cosX) ,
r\ + x\
the power consumed by the secondary resistance is,
_ 2 _ **ri (sin2 x + ^ cos2 x)
hence the difference, or the mechanical power developed by
the motor armature,
COMMUTATOR MOTORS. 361
and substituting for e,
egk cos X (x^ sin X + r^k cos X)
~ fa + r sin X + kx cos X)2 + (xl + x sin \ — kr cos X)2 '
and the torque in synchronous watts,
P <?02 cos X (x1 sin X + r^k cos X)
~~ /£ ~~ (/i + ?" sin A + £# cos X)2 + (xt + x sin X — kr cos X)2
or T= V27r^lO-8 [/!<!> sin X 7X cos A]' = [^/! cos X}>
_ ^ cos X (xl sin X + r^k cos X)
r2 + x2
The stationary torque is, k = 0,
_ ifo2^ sin X cos X
0 = (rx + r sin X)2 + (^ + * sin X)2 '
and neglecting the primary impedance, r = 0 = x,
_ e^x^ sin X cos X _ (fo2^ sin2 X
which is a maximum at X = 45°.
At speed k, neglecting r = 0 = x,
<?02 cos X (X sin X + r^k cos X)
— r2 j-^2 — ~'
which is a maximum for - — = 0, which gives,
cot 2 X = — . For k = 0, X = 45° ; for k = oo , X = 0.
that is, in the repulsion motor, with increasing speed, the
angle of secondary closed circuit, X, has to be reduced to
get maximum torque.
219. At A = 45° we have,
(rx V2 + r + £*)2 + (^ V2 + x - krf
and the power,
p= ^k (x, + r,K)_
(r, V2 + r + kx)*+(xi ^2 + x - krf'
362 ALTERNATING-CURRENT PHENOMENA.
this is a maximum, at constant X = 45°, for — — = 0, which
dk
gives, k = 1
At X = 0 we have,
T--
fa + kxf + (*t - krf
that is, T = 0 at k = 0, or, the motor is not self-starting,
when X = 0.
P =
dP
which is a maximum at constant X = 0 for, -— = 0, which
dk
gives,
rx-, — xr-.
MOO
--
'..i i'j
S
•^
"~
m
-t^>
,
/
no
1
/
/
R
:PL
LS
ON
M(
5TC
3R
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m
0
/
V
OC
rt
/
/
r=
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r, '
05
joa
>
/
X
2.
x.
1.
M
/
p-
DO
1.17
0 1
j^ ^
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g
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-,]
I
21 K I
/
UW
K1
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s
2.
I)
F/fir. 161. Repulsion Motor.
As an instance is shown, in Fig. 161, the power output
as ordinates, with the speed k = N^_ / N as abscissae, of a
repulsion motor of the constants,
X = 45° e0 = 100.
r= .1 r1= .05
* = 2.0 *x = 1.0
giving the power,
10,000 f .02 + 1.41 k — .05 ffj
~~ .171 + 2 y&)2 + (3.14 - .1 Kf '
COMMUTATOR MOTORS.
SERIES MOTOR. SHUNT MOTOR.
220. If, in a continuous-current motor, series motor as
well as shunt motor, the current is reversed, the direction
of rotation remains the same, since field magnetism and
armature current have reversed their sign, and their prod-
Fig. 162. Series Motor.
net, the torque, thus maintained the same sign. There-
fore such a motor, when supplied by an alternating current,
will operate also, provided that the reversals in field and
in armature take place simultaneously. In the series motor
this is necessarily the case, the same current passing through
field and through armature.
With an alternating current in the field, obviously the
364 ALTERNATING-CURRENT PHENOMENA.
magnetic circuit has to be laminated to exclude eddy cur-
rents.
Let, in a series 'motor, Fig. 146,
<l> = effective magnetism per pole,
n = number of field turns per pole in series,
«i = number of armature turns in series between brushes,
/ = number of poles,
(R. = magnetic reluctance of field circuit,*
(R! = magnetic reluctance of armature circuit,!
4>i = effective magnetic flux produced by armature current
(cross magnetization) per pole,
r = resistance of field (effective resistance, including hys-
teresis),
rj = resistance of armature (effective resistance, including hys-
teresis),
N = frequency of alternations,
N± = speed in cycles per second.
It is then,
E.M.F. induced in armature conductors by their rotation
through the magnetic field (counter E.M.F. of motor).
E =4
E.M.F. of self-induction of field,
E' =
E.M.F. of self-induction of armature,
^/ = 27r«1^V<I>110-8,
E.M.F. consumed by resistance,
Er = (r + *i) I,
where
/ = current passing through motor, in amperes effective.
Further, it is :
Field magnetism : $ = n 7108 / (R
* That is, the main magnetic circuit of the motor.
t That is, the magnetic circuit of the cross magnetization, produced by the armature
reaction.
COMMUTATOR MOTORS. 365
Armature magnetism :
Wj/108
1 = "V";
Substituting these values,
(R
ptfNI
E' =
(R
E1 = ^^niNI .
Er = (r + rj) /
Thus the impressed E.M.F.,
or, since
i,2
x = 2 TT N^- = reactance of field ;
(R
2-n-jV— = reactance of armature
fti
and
/
« • «,
366 AL TERNA TING-CURRENT PHENOMENA.
221. The power output at armature shaft is,
J>= El
\ (R
(R
fi- *Ef
7T « 7V^
/2 n± N± x _j_ r _^_
The displacement of phase between current and E.M.F.
tan CD =
Neglecting, as approximation, the resistances r + rlf it
1 + |!
lan W = ? «j ^
7T /« 7V
^n2
1+^'
^
/« TV
COMMUTATOR MOTORS. 367
hence a maximum for,
3r
7T
substituting this in tan w, it is :
tan o> = 1, or, w = 45°.
222. Instance of such an alternating-current motor,
^ = 100 AT=60 p = 2.
r = .03 ri = .12
x = .9 *! = .5
n = 10 »j = 48